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America, Labs NEC i w rhnra bases orthonormal Two {| πi/p needneWay Independence 4 x = Wotr n ils[4)Let [24]) Fields and (Wootters v , b atnR¨otteler Martin UTUALLY a,b 4 q nteseta eopsto.Ivanovi´c decomposition. spectral the in , − ) | i 1 n .I sa pnpolmwhether problem open an is It 5. and hntesadr ai oehrwith together basis standard the Then . / b 2 2 ( ∈ tr( ω × C |h p tr( F d d ρ d 2 U b q q ax | } | 1 + . 6 = ≥ c b NBIASED est arx atappar- fact a matrix, density , i| ih 2 a + 2 b 2 bx ∈ 1 = | hr r eea con- several are There . selusive. is uulyubae bases unbiased mutually = ) ) F ) x q /d O ∈ omastof set a form , h B F B b q = | od o all for holds ASES and ρ ∈ ρ P | b C fa ensemble an of i b C q ∈ skonfor known is q , B d of ea odd an be x +1 b C | b b d q d ih non- ∈ 1 + 1 + σ are b B x | d e , , Construction II (Galois Rings [13]) Let GR(4,n) be a finite Then we have that: Galois ring with Teichm¨uller set n. Define • M(pr)= pr +1 for p prime, r N, T ∈ 2πi • M(n) n +1 for all n N, v =2−n/2 exp tr(a +2b)x . ≤ ∈ N a,b   4  • M(mn) min M(m),M(n) for all m,n . | i x∈Tn ≥ { } ∈ • M(d2) N(d), where N(d) is the number of mutually ≥ Then the standard basis together with the bases Ma = orthogonal Latin squares of size d d. n va,b b n , a n, form a set of 2 + 1 mutually × n An open problem is to show that lim inf n→∞ M(n)= . unbiased{| i | bases∈ T of} C2∈. T ∞ Construction III (Bandyopadhyay et al. [1]) Suppose there III. WELCH’S LOWER BOUNDS exist subsets 1,..., m of a unitary error basis such that Suppose that X is a finite nonempty set of vectors of unit C C B Cd i = d, i j = 1d for i = j, and the elements of i norm in the complex vector space . The vectors in X satisfy pairwise|C | commute.C ∩C Let{ M} be a matrix6 which diagonalizes C. the inequalities i Ci Then M1,...,Mm are MUBs. 1 1 x y 2k , (1) Construction IV (Wocjan and Beth [23]) Suppose there are X 2 |h | i| ≥ d+k−1 | | x,yX∈X k w mutually orthogonal Latin squares [4], each of size d d
over the symbol set S = 1,...,d . Then w + 2 MUBs× for all integers k 0. Welch derived these bounds in [22] ≥ in dimension d2 can be constructed{ } as follows. With each to obtain a lower bound on the maximal cross-correlation of Latin square L (and additionally the square (1, 2,...,n)t spreading sequences of synchronous code-division multiple- (1,..., 1)) we can associate vectors of length d over the⊗ access systems. Blichfeld [5] and Sidelnikov [20] derived alphabet 1,...,d2 : for each symbol α S define a vector similar bounds for real vectors of unit norm. {d } ∈ sL,α C as follows: start with the empty list sL,α = . A set X attaining the Welch bound (1) for k = 1 is Then∈ traverse the elements of L column-wise starting at the∅ called a WBE-sequence set, a notion popularized by Massey upper left corner. Whenever α occurs in position (i, j) in L, and Mittelholzer [15] and others. Using equation (1), it is then append the number i + jd to the list sL,α. The other straightforward to check that the union of d + 1 mutually ingredient to construct these MUBs is an arbitrary complex unbiased bases of Cd form a WBE-sequence set. These Hadamard matrix H = (hi,j ) of size d d. For each Latin extremal sets of mutually unbiased bases are even better, since square L and each α, j 1,...,d define× a normalized they also attain the Welch bound for k = 2. In fact, we √ ∈d { } show that a sequence set attains the Welch bounds (1) for all vector vL,α,j := 1/ d i=1 esL,α[i]hi,j , where ei are the | i 2 elementary basis vectorsP in Cd . Then the bases given by k t if and only if it is a t-design in the complex projective space≤ CP d−1. BL := vL,α,j : α, j = 1,...,d together with the identity {| i } Let us introduce some notation. Let Sd−1 denote the sphere matrix 1d2 form a set of w +2 MUBs. of unit vectors in the complex vector space Cd. We say that Example 1: In dimension d = 3 Construction I yields the two vectors u and v of Sd−1 are equivalent, in signs u bases v, if and only if u = eiθv for some θ R. It is easy to≡ −1/2 2 2 ∈ 3 (1, 1, 1), (1,ω3,ω3), (1,ω3,ω3) , see that is an equivalence relation. We denote the quotient { } ≡ − − − −1/2 2 2 manifold Sd 1/ by CSd 1. Notice that the manifold CSd 1 3 (1,ω3,ω3), (1,ω3, 1), (1, 1,ω3) , { } is isomorphic to≡ the complex projective space CP d−1, but we 3−1/2 (1,ω2,ω2), (1,ω , 1), (1, 1,ω ) , { 3 3 3 3 } prefer the former notation because normalizing vectors to unit which together with the standard basis 13 form a maximal length is common practice in quantum computing. system of four MUBs in C3. Lemma 1: Let µ be the unique normalized U(d)-invariant Example 2: In dimension d =4 Construction II yields the Haar measure on the complex sphere CSd−1. For any x bases (where we have abbreviated “+” for 1 and “ ” for Sd−1, we have ∈ 1 and i = √ 1): − − − 2k 1 1 x y dµ(y)= d+k−1 . (+, +, +, +), (+, +, , ), (+, , , +), (+, , +, ) , ZC d−1 |h | i| 2 { − − − − − − } S k 1 Proof: The unitary group U(d) acts transitively on the 2 (+, , i, i), (+, ,i,i), (+, +, i, i), (+, +, i, i) ,
{ − − − − − − } manifold CSd−1. This means that for any y CSd−1 there 1 (+, i, i, ), (+, i, i, +), (+, i, i, ), (+, i, i, +) , ∈ 2 { − − − − − − } exists a unitary matrix U mapping y to the first basis vector, 1 (+, i, , i), (+, i, +,i), (+, i, +, i), (+, i, ,i) . Uy = e . Therefore, 2 { − − − − − − } 1 These four bases and the standard basis 14 form an extremal 2k 2k C4 x y dµ(x) = Ux e1 dµ(x) set of five MUBs in . ZCSd−1 |h | i| ZCSd−1 |h | i| A basic question is how many bases can be achieved in 2k = x e1 dµ(x), general dimension. To this end, we define the function M : −1 ZCSd |h | i| N N as follows: → where the last equality holds because of the U(d)-invariance M(n) := max : is a set of MUBs in Cn of the measure µ. Using Proposition 1.4.9 from Rudin [17], {|B| B } we obtain We show next that 2) implies 3). We observe that (2) holds for all k t, hence summing over x X yields (3). 2k k 2 1 ≤ ∈ x e1 dµ(x)= x1 dµ(x)= d+k−1 , d−1 d−1 Finally, we show that 3) implies 1). Suppose that equation ZCS |h | i| ZCS | | − d 1 d ⊗k
(3) holds. For a vector x C , we denote by x the k- ⊗ ∈ k which proves the claim. fold tensor product x k = x x Cd . Note that x⊗k y⊗k = x y k. Consider⊗···⊗ the d2k-dimensional∈ vector IV. COMPLEX PROJECTIVE t-DESIGNS h | i h | i 1 We now present some background material on complex ξ = x⊗k x⊗k x⊗k x⊗kdµ(x). projective designs. We will relate those later on to the systems X ⊗ − ZC d−1 ⊗ | | xX∈X S of vectors formed by a maximal set of MUBs. Let us first introduce some notation. We denote Evaluating the inner product of ξ with itself yields by Hom(k,ℓ) the subset of the polynomial ring 1 C 2k 2k (4) [x1,...,xd,y1,...,yd] that consists of all polynomials that 2 x y x y dµ(y)dµ(x), C d−1 X ∈ |h | i| − Z Z S |h | i| are homogeneous of degree k in the variables x1,...,xd | | x,yXX and homogeneous of degree ℓ in the variables y1,...,yd. which is equal to ξ ξ 0. The inner integral evaluates to We associate to each polynomial p in Hom(k,ℓ) a function d+k−1 −1 h | i ≥ d−1 by Lemma 1, and the double integral has the same p◦ on the sphere S by defining p◦(ξ) = p(ξ, ξ) for k d−1 value, because
the measure µ is normalized. It follows from ξ S . The function p◦ is called the “restriction” of p our assumption that the right hand side vanishes. By con- onto∈ the complex sphere. It follows from the homogeneity iϑ iϑ(k−ℓ) struction of ξ, we can conclude that averaging over X yields conditions of the polynomial p that p◦(e ξ)= e p◦(ξ) an exact cubature formula for all monomials in Hom(k, k)◦, holds for all ϑ R. Therefore, we obtain a well-defined hence, by linearity, for all polynomials in Hom(k, k)◦. This polynomial function∈ on CSd−1 only if k = ℓ. We define means that X is a t-design. Hom(k, k)◦ = p◦ : p Hom(k, k) . { ∈ } Remark 1: Equation (4) provides a short proof of the Welch Definition 2: A finite nonempty subset X of CSd−1 is a − inequalities (1). The analogue for real spherical t-designs t-design in CSd 1 iff the cubature formula of the above result is sketched in [7]. A connection to the 1 1 existence of certain isometric Banach space embeddings is f(x)= d−1 f(x)dµ(x) X µ(CS ) ZC d−1 given in [14]. | | xX∈X S holds for all f in Hom(t,t)◦. V. UNIFORM TIGHT FRAMES We now show a characterization of t-designs in terms of A finite subset F of nonzero vectors of Cd is called a frame the inequalities by Welch given in equation (1). if there exist nonzero real constants A and B such that Theorem 1: Suppose that X is a finite nonempty subset of 2 2 2 C d−1. Then the following statements are equivalent: A v f v B v S k k ≤ |h | i| ≤ k k fX∈F 1) The set X is a t-design in CSd−1; 2) for all x Cd and all k in the range 0 k t, we have Cd ∈ ≤ ≤ holds for all v . The notion of a frame generalizes the the equality concept of an orthonormal∈ basis. The linear span of the vectors Cd x x k 1 in F the space , but the vectors in a frame are in general h | i = x y 2k; (2) not linearly independent. A frame is called tight if and only if d+k−1 X |h | i| k | | yX∈X the frame bounds A and B are equal. A tight frame is called
3) the set X satisfies the Welch bounds (1) with equality for isometric (or uniform) if and only if each vector in F has unit all k in the range 0 k t, that is norm. ≤ ≤ Theorem 2: Let F be a finite nonempty subset of vectors 1 1 x y 2k = , 0 k t. (3) in Cd. The following statements about F are equivalent: X 2 |h | i| d+k−1 ≤ ≤ | | x,yX∈X k 1) F is a uniform tight frame; Proof: We show that 1) implies 2).
Fix a vector x Cd. ∈ 2) F is a WBE-sequence set; Note that p(y) = x y 2k = y x k x y k is a polynomial C d−1 |h | i| h | i h | i 3) F is a 1-design in S . function in Hom(k, k)◦. Since X is a t-design, the exact Proof: The frame constants of a uniform tight frame F cubature formula in Cd are given by A = B = F /d, see for example Property 1 2.3 in [8]. Therefore, F satisfies| | equation (2) of Theorem 1 x y 2k = x y 2kdµ(y) X |h | i| ZC d−1 |h | i| for k =1. The equivalence of the three statements follow now yX∈X S | | from Theorem 1. holds for all k in the range 0 k t. By Lemma 1, the latter Corollary 1: Any 1-design in CP d−1 is obtained by pro- d+k−1 ≤−1 ≤ integral evaluates to k , which proves that equation jecting an orthogonal basis from a higher-dimensional space (2) holds for all k t.
(where all basis vectors have the same norm). ≤ VI. EQUIVALENCE OF MUBS AND 2-DESIGNS We can now evaluate the intersection polynomials gij (0) using [11, Theorem 5.3] and obtain that I(x, y) = d 2. We need a few more notations before we state our main | | − results. If is a subset of CSd−1, then the set A = Hence we can conclude that each set Bx forms an orthonor- 2 B mal basis of Cd. The sets B partition . If B = B , then x y : x, y , x = y is called the “angle” set of . x B x 6 z For{|h | ani| element ∈x Bin 6and} an “angle” α A, we define theB the bases are by construction mutually unbiased. B ∈ 2 Zauner conjectures that if the dimension d is not a prime subdegree dα(x) as dα(x) = y : x y = α . If the |{ ∈ B |h | i| }| power, then a 2-design with angle set 0, 1/d cannot have subdegree dα of an α A is independent of x, then is called { } a regular scheme. Note∈ that the union of mutually orthogonalB d(d+1) elements [25]. His conjecture can now be reformulated bases of Cd is a regular scheme with angle set 0, 1/d . in terms of mutually unbiased bases, which then states that { } N(d) < d +1 for non-prime power d. If Zauner’s conjecture Theorem 3: The union X of d +1 mutually unbiased bases is true, then this would explain the particular role of the finite in Cd forms a 2-design in CSd−1 with angle set 0, 1/d and field construction by Wootters and Fields [24]. d(d + 1) elements. { } Remark 2: Theorem 3 was obtained earlier by Zauner as Proof: We verify that X attains the Welch bound in part of a more general result on combinatorial quantum designs equation (1) with equality for 0 k 2. The statement then ≤ ≤ using a different terminology, see [25, Theorem 2.19]. The follows from Theorem 1. Indeed, this is obvious for k = 0. converse direction, our Theorem 4, appears to be new. We note that X = d(d + 1). If we evaluate| | the left hand side of the Welch bound for X, VII. SIC-POVMS AND 2-DESIGNS then we obtain Finally, to demonstrate the versatility of Theorem 1 we also 1 d(d+1) 1 show that another system of vectors used in quantum informa- x y 2 = 1+(d 1)0+d2 d2(d+1)2 |h | i| d2(d+1)2  − d tion theory corresponds to complex projective 2-designs. So- x,yX∈X called symmetric informationally complete positive operator- 1 2 = valued measures (SIC-POVMs) are systems of d vectors d in Cd which have constant inner product, i.e., v, w 2 = −1 |h i| and this coincides with d+1−1 =1/d; so, X is a 1-design. 1/(d + 1) for all v, w in the set. Like in case of MUBs it 1 is a challenging task to construct SIC-POVMs—indeed here Similarly, for k =2,
solutions are known only for a finite number of dimensions 1 d(d+1) 1 x y 4 = 1+(d 1)0+d2 [9], [16]. In [16] it was shown that SIC-POVMs actually form d2(d+1)2 |h | i| d2(d+1)2  − d2  x,yX∈X complex projective 2-designs. The following theorem gives a 2 new and simple proof of this result. = , d(d + 1) Theorem 5 (SIC-POVMs are 2-designs [16]): Let X be a SIC-POVM X in dimension d. Then X forms a 2-design in d+2−1 −1 and this coincides with =2/(d(d + 1)). CSd−1 with angle set 1/(d + 1) and d2 elements. 2 { } Theorem 4: A 2-design
in complex projective space Proof: Again, we only have to verify that the set X of − B CSd 1 with angle set 0, 1/d and = d(d + 1) elements vectors attains the Welch bound with equality for 0 k 2. { } |B| is the union of d +1 mutually unbiased bases. The statement then follows from Theorem 1. Indeed,≤ this≤ is Proof: A complex projective 2-design with s = obvious for k = 0. We note that here X = d2. Evaluating | | 0, 1/d =2 satisfies 2 s 1, hence is a regular scheme the left hand side of the Welch bound for X, then we obtain |{ }| ≥ − 2 [10]. For α = 1/d, any x has subdegree dα(x) = d by 1 1 1 ∈B x y 2 = d2 1+(d4 d2) Theorem 2.5 of [10]. It follows that x is orthogonal to d 1 d4 |h | i| d4  · − d +1 elements. − x,yX∈X 1 1 Let Bx = x z : x z =0 . We claim that Bx is = (1 + (d 1)) = an orthonormal{ }∪{ basis of∈BCd.h We| i may assume} that the vectors d2 − d in are normalized to unit norm. Thus, it suffices to show that d+1−1 −1 B and this coincides with 1 =1/d; so, X is a 1-design. Bx = By for each y Bx. For x = y this is trivial. We know Similarly, for k =2,
that x and y are contained∈ in both B and B . Therefore, it x y 1 1 1 suffices to show that the intersection set x y 4 = d2 1 + (d4 d2) d4 |h | i| d4  · − (d + 1)2  x,yX∈X I(x, y)= z : x z =0, y z =0 = Bx By x, y { ∈B h | i h | i } ∩ −{ } 1 d 1 2 = 1+ − = contains d 2 elements. d2  d +1 d(d + 1) The number− of elements in I(x, y) does not depend on −1 x, y for a t-design with t 2s 2, see [11]. Specializing and this coincides with d+2−1 =2/(d(d + 1)). ≥ − 2 Theorem 5.2 in [11] to the case at hand shows that Remark 3: Zauner pointed out
to us that the previous the- 1 orem can also be obtained in the language of combinatorial I(x, y) = d2 σ0 σ0 d(d + 1)g (0) 0i 0j . quantum designs by combining Theorems 2.29 and 2.30 in his | | 1−i 1−j ij − − i,jX=0
dissertation [25]. VIII. CONCLUSION [4] Th. Beth, D. Jungnickel, and H. Lenz. Design Theory, volume I. Cambridge University Press, 2. edition, 1999. We have shown that the seemingly unrelated concepts of [5] H.F. Blichfeld. The minimum value of quadratic forms, and the closest MUBs on the one hand and complex projective 2-deigns on packing of spheres. Math. Ann., 101:605–608, 1929. [6] N. Bohr. Das Quantenpostulat und die neueren Entwicklungen der the other are actually the same objects. This was anticipated Atomistik. Naturwissenschaften, 16:245–257, 1928. in a paper by Barnum [2] in which it was shown that the [7] J.M. Goethals and J.J. Seidel. Cubature formulae, polytopes, and union of the (d+1) bases of a particular system of MUBs spherical designs. In C. Davis, B. Gr¨unbaum, and F.A. Sherk, editors, The Geometric Vein – The Coxeter Festschrift, pages 203–218. Springer- forms a complex projective 2-design. In the present paper we Verlag, 1981. have generalized this to arbitrary MUBs and have also shown [8] V.K. Goyal, J. Kovaˇceviˇc, and J.A. Kelner. Quantized frame expansions the other direction, i.e., any 2-design in dimension d which with erasures. Appl. Comp. Harm. Analysis, 10:203–233, 2001. [9] M. Grassl. On SIC-POVMs and MUBs in Dimension 6. pages 60– consists of d2 + d elements and has angle set 0, 1/d can be { } 61. Proceedings ERATO Conference on Quantum Information Science partitioned into d +1 sets of MUBs. We have also shown that (EQIS 2004), Tokyo, 2004. See also ArXiv preprint quant-ph/0406175. −1 these sets meet the Welsh bounds for k =0, 1, 2 with equality. [10] S.G. Hoggar. Parameters of t-designs in FP n . Europ. J. Combin., 5:29–36, 1984. Hence, the present paper can also be seen as a generalization [11] S.G. Hoggar. t-designs with general angle set. Europ. J. 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Welch’s bound and sequences sets to our attention, and to Gerhard Zauner for assisting with for code-division multiple-access systems. In Sequences II: Methods the translation of his results. We also thank Chris Godsil, in Communications, Security and Computer Sciences, pages 63–78, Markus Grassl, Joe Renes, Aidan Roy for several interesting Heidelberg, 1993. Springer-Verlag. [16] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves. Sym- discussions, and Richard Cleve for pointing out a small error metric informationally complete quantum measurements. J. Math. Phys, in an example of a previous version. 45(6):2171–2180, 2004. The research of A.K. was supported in part by NSF grant [17] W. Rudin. Function Theory in the Unit Ball of Cn. Springer-Verlag, New York, 1980. CCR-0218582, NSF CAREER award CCF-0347310, a TEES [18] J. Schwinger. Unitary operator bases. Proc. Nat. Acad. Sci. U.S.A., Select Young Faculty award, and a Texas A&M TITF grant. 46:570–579, 1960. 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